The most important and trickiest part of cat genetics is getting a handle on the sex-linked red gene, especially if you want to learn to predict the offspring of two cats. Because this gene is located on the X chromosome, it behaves quite differently from the other genes we'll be looking at.
This is also the reason you might have heard that male kittens take after their mother, while mollies take after both parents when it comes to pelt color. While true in a sense, I strongly advise you to forget it, since it seems to confuse as many people as it helps.
In cats and most other mammals, there are two main types of pigment in the skin and hair. Eumelanin is the black pigment, while phaeomelanin is the red pigment. In cats, eumelanin is responsible for black and brown colors, while phaeomelanin is responsible for red, orange, and yellow tones.
Your typical plain black cat actually has both eumelanin and phaeomelanin in its fur, but because eumelanin is so much darker, it becomes the dominant effect, and you can't see the phaeomelanin's color, except sometimes as a slight reddish cast in the sunlight. The sex-linked red gene simply prevents the cat from producing eumelanin, allowing the phaeomelanin's color to show.
The tricky bit is that, as previously mentioned, this gene is on the X chromosome. In male cats with only one X chromosome, this is simple - either they are red, or they aren't. But female cats have two X chromosomes to work with, and so they can be tortoiseshell, having patches of both red and black. This happens when one X chromosome has the red allele (XO) and the other has the "not-red" allele (Xo). (The more proper notation is Xᴼ and Xᵒ, but those are annoyingly difficult to read.) The O is for orange, though as I'm using breeder's terms, I'll continue to call it red.
If we want to predict how the Red gene will pass on, we need to do a bit of trickery with our Punnett squares. In this case, the mother can pass on either of her two X chromosomes at random, and the father will pass on either his X or his Y chromosome. Kittens that receive the father's X chromosome will be XX, and be female, while those that receive a Y chromosome will be XY, and be male.
The important thing to realize here is that male kittens get their X chromosome, and therefore, their O gene, from their mother, while female kittens get one copy of the gene from each parent. So, in a sense, the saying from the start is correct.
However, this only applies to the sex-linked red gene. If we are talking about two eumelanin-based ("not-red") colors, the kittens will not inherit colors based on sex. And if we are dealing with both sex-linked red and autosomal genes (genes that aren't on the sex chromosomes), we need to remember to separate those factors accordingly.
What about tortoiseshell toms?
By this model, male cats can't be tortoiseshells. Usually, this is true, but there are rare cases of just that. There are two mechanics by which a tom can be tortoiseshell.
The first is an intersex condition known as Klinefelter Syndrome. In this case, due to an error in gamete production, the cat ends up being XXY. Because of the Y chromosome, the cat appears to be male, but because of the two X chromosomes, he may be tortoiseshell. These cats are infertile, since the chromosome weirdness messes with gamete formation. Other intersex conditions can also lead to tortoiseshell toms, but this seems to be the most common.
The other, much less common option, is that the cat be a chimera. Chimeras, in the genetics sense, are a fusion of two separate embryos. They appear to be one individual, but genetically speaking, they are a mix of two different individuals. These cats can be fertile, but will breed as though they are either red or black.
For more information on tortoiseshell toms, Messybeast has several articles on the topic.
I made this little visualizer so you can get a feel for how the genes interact as we go. This version is pretty simple, but we’ll build it up over time! (I know, embedding the JSFiddle isn’t ideal, but Squarespace is being uncooperative with jQuery.)
I'll end this with a gallery of Punnett squares for each possible pairing, so you can reference them if you need. Feel free to save and share them as you please.
OMIA - Coat color, orange in Felis catus. Pretty much every paper that's ever been written on the subject is included here. Most of the more recent studies are examinations of tortoiseshell toms.
Messybeast - Basic genetics for breeders and cat lovers. Same link from earlier, lots of info on tortoiseshell toms if that's something you're interested in!